Review

Use these notes to review Chapter 4 quickly.

Key Ideas

A point is a location in a space.
A direction is a movement or change.
A coordinate system is a convention for naming points and directions.
Length measures the size of one vector.
Distance measures the length of the difference between two points.
Angle compares two directions.
Orthogonal means perpendicular; the dot product is zero.
Projection keeps a component of one vector along another direction.
A basis gives directions for describing vectors in a space.
A subspace is a smaller space closed under addition and scaling.
A hyperplane is a boundary written as $w \cdot x + b = 0$ .
An embedding is a learned or computed vector representation of an object.
A high-dimensional view is a projection or summary of a larger space.

Formulas to Remember

Euclidean length in two dimensions:

\|[x, y]\| = \sqrt{x^2 + y^2}

Euclidean length in (d) dimensions:

\|x\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_d^2}

Distance:

d(a, b) = \|b - a\|

Dot product and angle:

u \cdot v = \|u\|\|v\|\cos(\theta)

Projection onto a nonzero vector:

\operatorname{proj}_b(a) = \frac{a \cdot b}{b \cdot b}b

Hyperplane:

w \cdot x + b = 0

Signed distance from a point to a hyperplane:

\frac{w \cdot x + b}{\|w\|}

Reading Guide

For coordinates, ask: point or direction?
For distance, ask: which metric defines close?
For a dot product, ask: score, angle, or projection?
For projection, ask: scalar component or projected vector?
For a residual, ask: what did the projection remove?
For a hyperplane, ask: score positive, negative, or zero?
For a boundary score, ask: raw score or normalized distance?
For an embedding, ask: evidence of structure, or overclaim?
For a projection plot, ask: what was kept, and what was discarded?
For high dimension, ask: does the picture need a formula, numerical example, or code check?

Checks Before You Move On

Treating a point and a direction as the same idea because the numbers match.
Forgetting that distance depends on the chosen metric.
Calling vectors orthogonal without checking the dot product.
Projecting onto a non-unit vector without dividing by $b \cdot b$ .
Confusing the boundary with the regions on either side.
Forgetting that a subspace must contain zero.
Treating nearby embeddings as complete explanations.
Trusting a two-dimensional picture too much in a high-dimensional setting.
Reading raw classifier score as distance without accounting for $\|w\|$ .
Forgetting that coordinates are weights on a chosen basis.
Treating a PCA, t-SNE, or UMAP plot as the full original space.

Mental Model

Geometry gives shape to computation.

When a formula uses vectors or matrices, ask whether it is measuring length, comparing directions, projecting onto a component, or separating space with a boundary. When a picture shows a high-dimensional object, ask what projection or summary produced the picture.